Extension of the principal-value representation of \(\mathsf{X}_1(w)\) to complex parameters

Establish whether the representation of the Hua–Pickrell random variable \(\mathsf{X}_1(w)\) as the almost-sure limit of the truncated difference between the positive and negative points of the determinantal point process \(\mathcal{C}^{(w)}\) remains valid for all nonreal parameters \(w\in\mathbb{C}\setminus\mathbb{R}\).

Background

For real parameters $w$ with $2w>-1$, Qiu’s result identifies $\mathsf{X}_1(w)$ almost surely with the limit of the truncated difference between the positive and negative points of the determinantal point process $\mathcal{C}^{(w)}$. This representation connects the first Hua–Pickrell ergodic parameter $\gamma_1$ with the point-process coordinates.

The paper explicitly states that the validity of this same representation is not known for nonreal parameters. Such an extension would provide a direct determinantal-point-process description of $\mathsf{X}_1(w)$ in the complex-parameter regime, which the paper otherwise studies through Painlevé and analytic methods.